In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut can also be viewed as a generalization of symplectic blow up. The cut was introduced in 1995 by Eugene Lerman, who used it to study the symplectic quotient and other operations on manifolds.
Topological description Let ( X , ω ) {\displaystyle (X,\omega )} be any symplectic manifold and
μ : X → R {\displaystyle \mu :X\to \mathbb {R} }
a Hamiltonian on X {\displaystyle X} . Let ϵ {\displaystyle \epsilon } be any regular value of μ {\displaystyle \mu } , so that the level set μ − 1 ( ϵ ) {\displaystyle \mu ^{-1}(\epsilon )} is a smooth manifold. Assume furthermore that μ − 1 ( ϵ ) {\displaystyle \mu ^{-1}(\epsilon )} is fibered in circles, each of which is an integral curve of the induced Hamiltonian vector field. Under these assumptions, μ − 1 ( [ ϵ , ∞ ) ) {\displaystyle \mu ^{-1}([\epsilon ,\infty ))} is a manifold with boundary μ − 1 ( ϵ ) {\displaystyle \mu ^{-1}(\epsilon )} , and one can form a manifold
X ¯ μ ≥ ϵ {\displaystyle {\overline {X}}_{\mu \geq \epsilon }}
by collapsing each circle fiber to a point. In other words, X ¯ μ ≥ ϵ {\displaystyle {\overline {X}}_{\mu \geq \epsilon }} is X {\displaystyle X} with the subset μ − 1 ( ( − ∞ , ϵ ) ) {\displaystyle \mu ^{-1}((-\infty ,\epsilon ))} removed and the boundary collapsed along each circle fiber. The quotient of the boundary is a submanifold of X ¯ μ ≥ ϵ {\displaystyle {\overline {X}}_{\mu \geq \epsilon }} of codimension two, denoted V {\displaystyle V} . Similarly, one may form from μ − 1 ( ( − ∞ , ϵ ] ) {\displaystyle \mu ^{-1}((-\infty ,\epsilon ])} a manifold X ¯ μ ≤ ϵ {\displaystyle {\overline {X}}_{\mu \leq \epsilon }} , which also contains a copy of V {\displaystyle V} . The symplectic cut is the pair of manifolds X ¯ μ ≤ ϵ {\displaystyle {\overline {X}}_{\mu \leq \epsilon }} and X ¯ μ ≥ ϵ {\displaystyle {\overline {X}}_{\mu \geq \epsilon }} . Sometimes it is useful to view the two halves of the symplectic cut as being joined along their shared submanifold V {\displaystyle V} to produce a singular space
X ¯ μ ≤ ϵ ∪ V X ¯ μ ≥ ϵ . {\displaystyle {\overline {X}}_{\mu \leq \epsilon }\cup _{V}{\overline {X}}_{\mu \geq \epsilon }.}
For example, this singular space is the central fiber in the symplectic sum regarded as a deformation.
Symplectic description The preceding description is rather crude; more care is required to keep track of the symplectic structure on the symplectic cut. For this, let ( X , ω ) {\displaystyle (X,\omega )} be any symplectic manifold. Assume that the circle group U ( 1 ) {\displaystyle U(1)} acts on X {\displaystyle X} in a Hamiltonian way with moment map
μ : X → R . {\displaystyle \mu :X\to \mathbb {R} .}
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